A circle C has centre M (6, 4) and radius 3 - Edexcel - A-Level Maths Pure - Question 9 - 2008 - Paper 2

To find the angle TMQ, we will use geometric properties. The coordinates of points T, P, and Q need to be determined from the given information. Once we ascertain these points, we can use the tangent properties or coordinate geometry methods to find the angle TMQ.

Using the tangent (slopes) and the relationship between the angles:

Let m1 be the tangent of angle T from point M to P, and m2 be the tangent of line M to Q:

Then use the formula:

an^{-1}rac{m_1 - m_2}{1 + m_1 m_2}

Compute the values to find the angle TMQ, confirming that it equals approximately 1.0766 radians.

The area of the shaded region TPQ can be calculated by integrating the area across the boundaries defined by segments TP and TQ.

Steps to find the area:

1. Identify the coordinates of points T, P, and Q.

2. Calculate the area of triangle formed by points T, P, and Q.

   Using the formula for the area of triangle: $ext{Area} = \frac{1}{2} | x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$

3. Calculate the arc length of the circle segment between points T and Q to find the remaining area.

4. Combine these areas to find the total shaded area TPQ, rounding the final result to three decimal places.